Geometry of quaternionic hyperbolic manifolds

Abstract

We develop some of the basic theory of quaternionic hyperbolic geometry. We give necessary criteria for groups of quaternionic hyperbolic motions to be discrete. We give lower bounds on the volumes of cusped quaternionic hyperbolic manifolds.

An extension problem for the CR fractional Laplacian

“…In the paper [52] the quaternionic hyperbolic space H m Q of quaternionic dimension m is characterized as a Siegel domain with boundary Q n ; we give here the main details of this construction. Consider Q m,1 , the quaternionic vector space of quaternionic dimension m + 1 (so real dimension 4m + 4) with the quaternionic Hermitian form given by .…”

Shimizu’s Lemma for Quaternionic Hyperbolic Space

“…and so Theorem 1.1, or Corollary 1.2, is just Theorem 4.8 of Kim-Parker [16]. If in addition τ = 0 then T (S −1 (∞)) = T (S(∞)) = |t| 1/2 , and we recover Kamiya [13,Thm.…”

“…Versions for isometry groups of H 2 C containing a loxodromic or elliptic map were given by Basmajian and Miner [1] and Jiang et al [9]. These results were extended to H 2 H by Kim and Parker [16] and Kim [15]. Cao and Parker [3] and Cao and Tan [4] obtained generalised Jørgensen's inequalities in H n H for groups containing a loxodromic or elliptic map.…”

“…Kamiya generalised Shimizu's lemma to subgroups of PU(n, 1) or PSp(n, 1) containing a vertical Heisenberg translation [13]. For subgroups of PU(n, 1) containing a general Heisenberg translation, Parker [20,21] gave versions of Shimizu's lemma both in terms of a bound on the radius of isometric spheres and a precisely invariant horoball or sub-horospherical region.This was generalised to PSp(n, 1) by Kim and Parker [16]. Versions of Shimizu's lemma for subgroups of PU(2, 1) containing a screw parabolic map were given by Jiang et al [10,14].…”

On the Classification of Four-Dimensional Möbius Transformations

“…Wada [10], Waterman [11], Cao and Waterman [2] and Kim and Parker [9] also had done some related research in this direction.…”